Category Archives: proof

Totient sums

I took a bit of a break to travel to Japan for a conference, but I’m back now to continue the series I started with Post Without Words #10, a follow-up post, and Post Without Words #11. Recall that we … Continue reading

Posted in geometry, pattern, pictures, posts without words, proof | Tagged , , , , , | 4 Comments

Post without words #11

Posted in geometry, pattern, pictures, posts without words, proof | Tagged , , , , , | 7 Comments

The route puzzle

While poking around some old files I came across this puzzle: (Click for a larger version.) I didn’t make it, and I have no idea where I got it from (do you know?). But in any case, wherever it comes … Continue reading

Posted in arithmetic, challenges, number theory, proof, puzzles | Tagged , , , , , , | 7 Comments

Golden numbers are Fibonacci

This post is fourth in a series, proving the curious fact that is a Fibonacci number if and only if one (or both) of or is a perfect square; we call numbers of this form golden numbers. Last time, I … Continue reading

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Fibonacci numbers are golden

Recall that a “golden number” (this is not standard terminology) is a number such that one (or both) of or is a perfect square. In this post, I’ll explain Gessel’s proof that every Fibonacci number is golden. First, we need … Continue reading

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Testing Fibonacci numbers: the proofs

In my last post I stated this surprising theorem: is a Fibonacci number if and only if one of is a perfect square. If one of is a perfect square, then let’s say that is a “golden number” (a nod, … Continue reading

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The chocolate bar game: losing positions proved

In my last post I claimed that the losing positions for the chocolate bar game are precisely those of the form (or the reverse), that is, in binary, positions where one coordinate is the same as the other with any … Continue reading

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